562 NOTES. 



NOTE II. 



ALGEBRA OF INDETERMINATE MULTIPLIERS. 



On page 61 we have an example of the use of indeterminate 

 multipliers in elimination. It may be somewhat more clear if we examine 

 in just what the process involved consists Equation 12) is a linear 

 equation involving the 3n quantities d^, ... dz n , each multiplied by a 

 coefficient which is independent of the d's. Besides this equation the 

 quantities d satisfy the equations 14), which are of the same form, that 

 is, linear in all the $'s, with coefficients independent of them. Aside 

 from this the d's may have any values whatever. It is for the purposes 

 of this discussion quite immaterial that the d's are small quantities, we 

 are concerned simply with a question of elimination. Let us accordingly 

 represent them by the letters x^ X 2 , ... # m , between which we have the 

 linear equation 



1) A^ 4- A 2 x 2 4- ---- h A m x m = 0. 



The x's are however not independent, but must in addition satisfy the 

 equations 



B^XI 4- B 12 x 2 - - - 4- BimXm = 0, 



B^ 4- #22^2 ---- H 



4- 



The number of these equations, &, is less than z, the number of the x's. 

 The question is now, what relations are involved among the A's and J5's 

 when the x's have any values whatever compatible with the equations 2). 



We may evidently proceed as follows. Transposing m k terms 

 in 2), say the last, we may solve the equations for the quantities 

 # 1? X 2 x ki as linear functions of the remaining #A_J_I, . . . x m . These 

 m k quantities are now perfectly arbitrary. Inserting the values of 

 x . . . Xk in equation l), this becomes linear in the m k quantities 

 afc+i, . . x m , which being purely arbitrary, in order for equation l) to 

 hold for all values of the x's, the coefficient of each must vanish, giving 

 us the required m k relations between the A's and B's. 



Instead of proceeding in the manner described, the method of 

 Lagrange is to multiply the equations 2) respectively by multipliers 

 Aj, yl 2 , . . A*, to which any convenient value may be given, and then 

 to add them to equation l). We thus obtain 



(A l 4- *! -B n 4- A 2 -B ai 4- 

 -f (A 2 + A! 5 12 + A 2 22 4- 



4- (44- *i-Blm+ AS ' ' 4- 



